CAT 2024 Slot 3 QA Q22 Logarithms & Exponents

Question 22 of the Quantitative Ability (QA) section, CAT 2024 Slot 3, with CAT's official answer and a worked solution.

The sum of all distinct real values of \(x\) that satisfy the equation \(10^x+\frac{4}{10^x}=\frac{17}{2}\), is

A
\(2 \log _{10} 2\)
B
\(3 \log _{10} 2\)
C
\(4 \log _{10} 2\)
D
\(\log _{10} 2\)

Answer: (A) 2 log _10 2 Logarithms & Exponents MCQ

Solution

Let \(y = 10^x\): \(y + \frac4y = \frac{17}{2} \Rightarrow 2y^2 - 17y + 8 = 0 \Rightarrow y = 8\) or \(\frac12\).
Sum of \(x\) \(= \log_{10}(8 \times \tfrac12) = \log_{10}4 = 2\log_{10}2\).

Shortcut. Product of the roots in \(y\) is \(\frac82 = 4\), so the sum of the \(x\)'s is \(\log_{10}4\) directly.